33:
103:
929:
weighted spaces, which is a formalization of the above observation. They were able to show that with this additional domain knowledge high dimensional integrals satisfying certain conditions were tractable even in the worst case! In contrast the Monte Carlo method gives only a stochastic assurance. See Sloan and Woźniakowski
547:
we have to settle for conjectures about the complexity hierarchy. The reason is that the input is a number or a vector of numbers and can thus be entered into the computer. Thus there is typically no adversary argument at the information level and the complexity of a discrete problem is rarely known.
208:
information. To give a simple illustration, in the numerical approximation of an integral, only samples of the integrand at a finite number of points are available. In the numerical solution of partial differential equations the functions specifying the boundary conditions and the coefficients of the
928:
dimensions in the CMO represent monthly future times. Due to the discounted value of money variables representing times for in the future are less important than the variables representing nearby times. Thus the integrals are non-isotropic. Sloan and Woźniakowski introduced the very powerful idea of
551:
The univariate integration problem was for illustration only. Significant for many applications is multivariate integration. The number of variables is in the hundreds or thousands. The number of variables may even be infinite; we then speak of path integration. The reason that integrals are
260:
Since the information is partial and contaminated, only approximate solutions can be obtained. IBC studies computational complexity and optimal algorithms for approximate solutions in various settings. Since the worst case setting often leads to negative results such as unsolvability and
216:
The goal of information-based complexity is to create a theory of computational complexity and optimal algorithms for problems with partial, contaminated and priced information, and to apply the results to answering questions in various disciplines. Examples of such disciplines include
696:(average case setting.) Another possibility is the randomized setting. For some problems we can break the curse of dimensionality by weakening the assurance; for others, we cannot. There is a large IBC literature on results in various settings; see Where to Learn More below.
203:
variable. Since one can never obtain a closed-form solution to the problems of interest one has to settle for a numerical solution. Since a function of a real or complex variable cannot be entered into a digital computer, the solution of continuous problems involves
974:
This annual prize, which was created in 1999, consists of $ 3000 and a plaque. It is given for outstanding achievement in information-based complexity. The recipients are listed below. The affiliation is as of the time of the award.
907:
These results are empirical; where does computational complexity come in? QMC is not a panacea for all high dimensional integrals. What is special about financial derivatives? Here's a possible explanation. The
1129:
This annual award, which was created in 1996, consists of $ 3000 ($ 4000 since 2015) and a plaque. Many, but by no means all of the awards have been for research in IBC. The recipients have been
1017:
2012 Michael
Gnewuch, Department of Computer Science, Christian-Albrechts-Universitaet zu Kiel, Germany and School of Mathematics and Statistics, University of New South Wales, Sydney, Australia
945:
had the key insight that the optimal algorithm and the computational complexity of solving a continuous problem depended on the available information. He applied this insight to the solution of
896:
beat MC by one to three orders of magnitude. The results were reported to a number of Wall Street finance to considerable initial skepticism. The results were first published by Paskov and
612:(Here we are counting the number of function evaluations and the number of arithmetic operations so this is the time complexity.) This would take many years for even modest values of
552:
important in many disciplines is that they occur when we want to know the expected behavior of a continuous process. See for example, the application to mathematical finance below.
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539:-approximation. Because of these information-based arguments we can often obtain tight bounds on the complexity of continuous problems. For discrete problems such as
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intractability, settings with weaker assurances such as average, probabilistic and randomized are also studied. A fairly new area of IBC research is
1472:
262:
209:
differential operator can only be sampled. Furthermore, this partial information can be expensive to obtain. Finally the information is often
1329:
Prentice Hall, 1964. Reissued
Chelsea Publishing Company, 1982; Russian translation MIR, 1985; Reissued American Mathematical Society, 1998
1071:
2004 Christiane
Lemieux, University of Calgary, Calgary, Alberta, Canada, and Josef Dick, University of New South Wales, Sydney, Australia
652:
occurs for almost every continuous problem that has been investigated. How can we try to vanquish the curse? There are two possibilities:
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1363:
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676:(worst case setting) and settle for a stochastic assurance. For example, we might only require that the expected error be less than
121:
84:
933:
J. Complexity 14, 1-33, 1998. For which classes of integrals is QMC superior to MC? This continues to be a major research problem.
112:
904:, Journal of Portfolio Management 22, 113-120. Today QMC is widely used in the financial sector to value financial derivatives.
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176:
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2013 Christoph
Aistleitner, Department of Analysis and Computational Number Theory, Technische Universitat Graz, Austria
242:
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188:
45:
893:
889:
55:
49:
41:
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2010 Boleslaw Z. Kacewicz, Department of
Mathematics, AGH University of Science and Technology, Cracow, Poland
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dimensions (dimensions and variables are used interchangeably) and that we want to guarantee an error at most
641:
285:
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Extensive bibliographies may be found in the monographs N (1988), TW (1980), TWW (1988) and TW (1998). The
1040:
2016 Fred J. Hickernell, Department of
Applied Mathematics, Illinois Institute of Technology, Chicago, USA
148:
66:
955:
The general setting for information-based complexity was formulated by Traub and Woźniakowski in 1980 in
999:
2006 Leszek
Plaskota, Department of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
949:, which started the area of optimal iteration theory. This research was published in the 1964 monograph
809:
712:
Very high dimensional integrals are common in finance. For example, computing expected cash flows for a
540:
192:
959:
For a list of more recent monographs and pointers to the extensive literature see To Learn More below.
349:
to compute the integral analytically; we have to approximate it numerically. We compute the values of
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for any integrand in some class. The computational complexity of the problem is known to be of order
164:
1301:
1077:
2006 Jakob
Creutzig, TU Darmstadt, Germany and Dirk Nuyens, Katholieke Universiteit, Leuven, Belgium
1065:
This annual award, which created in 2003, consists of $ 1000 and a plaque. The recipients have been
1008:
2009 Thomas
Mueller-Gronbach, Fakultaet fuer Informatik und Mathematik, Universitaet Passau, Germany
779:
1388:
The
Computational Complexity of Differential and Integral Equations: An Information-Based Approach,
877:
238:
504:
numbers by a combinatory algorithm to compute an approximation to the integral. See the monograph
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2005 Friedrich Pillichshammer, Institute for Financial Mathematics, University of Linz, Austria
679:
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2012 (Special prize) Krzysztof Sikorski, Department of Computer Science, University of Utah
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2003 Frances Kuo, School of Mathematics, University of New South Wales, Sydney, Australia
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2014 Frances Kuo, School of Mathematics, University of New South Wales, Sydney, Australia
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2004 Peter Mathe, Weierstrass Institute for Applied Analysis and Stochastics, Germany
648:
We stated the curse of dimensionality for integration. But exponential dependence on
273:
We illustrate some important concepts with a very simple example, the computation of
250:
1037:
2015 Peter Kritzer, Department of Financial Mathematics, University of Linz, Austria
996:
2005 Ian Sloan, Scientia Professor, University of New South Wales, Sydney, Australia
941:
Precursors to IBC may be found in the 1950s by Kiefer, Sard, and Nikolskij. In 1959
1121:
2018 Arnulf Jentzen, Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland
1104:
2015 Mario Ullrich, Institute of Analysis, Johannes Kepler University Linz, Austria
1054:
2018 Paweł Przybyłowicz, AGH University of Science and Technology in Kraków, Poland
946:
254:
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2014 Tino Ullrich, Institute for Numerical Simulation, University of Bonn, Germany
1095:
2012 Pawel Przybylowicz, AGH University of Science and Technology, Cracow, Poland
931:
When are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integration?
196:
160:
1014:
2011 Aicke Hinrichs, Fakultät für Mathematik und Informatik, FSU Jena, Germany
876:(MC), an instance of a randomized algorithm. Then in 1994 a research group at
776:
years. Recall that if a worst case assurance is required the time is of order
246:
230:
144:
1219:
2009 Frank Aurzada, Steffen Dereich, Michael Scheutzow and Christian Vormoor
156:
1350:
Lecture Notes in Mathematics, vol. 1349, Springer-Verlag, New York, 1988
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218:
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spaces, while the applications are usually for multivariate problems.
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885:
195:. All these problems involve functions (typically multivariate) of a
1057:
2019 Jan Vybíral, Czech Technical University, Prague, Czech Republic
1029:
Friedrich Pillichshammer, Johannes Kepler University, Linz, Austria
1002:
2007 Klaus Ritter, Department of Mathematics, TU Darmstadt, Germany
881:
1458:
J.F Traub, 1985. An Introduction to Information-Based Complexity
1348:
Deterministic and Stochastic Error Bounds in Numerical Analysis,
1354:
Traub, J. F., Woźniakowski, H., and Wasilkowski, G. W. (1988).
1269:
2015 Thomas Müller-Gronbach, Klaus Ritter, Larisa Yaroslavtseva
1430:
96:
26:
981:
2000 Sergei Pereverzev, Ukrainian Academy of Science, Ukraine
656:
We can weaken the guarantee that the error must be less than
468:
numbers are the partial information about the true integrand
1026:
Josef Dick, University of New South Wales, Sydney, Australia
987:
2002 Stefan Heinrich, University of Kaiserslautern, Germany
1211:
Istvan Berkes, Robert F. Tichy and the late Walter Philipp
249:. The theory is developed over abstract spaces, typically
1339:
Traub, J. F., Woźniakowski, H., and Wasilkowski, G. W.,
1228:
Simon Foucart, Alain Pajor, Holger Rauhut, Tino Ullrich
1080:
2007 Andreas Neuenkirch, Universität Frankfurt, Germany
872:
time units. People in finance have long been using the
511:
Because we have only partial information we can use an
1275:
David Harvey, Joris van der Hoeven and Grégoire Lecerf
1005:
2008 Anargyros Papageorgiou, Columbia University, USA
972:
Prize for Achievement in Information-Based Complexity
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Carlos Beltrán, Jordi Marzo and Joaquim Ortega-Cerdà
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Information-Based Complexity Young Researcher Award
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2001 G. W. Wasilkowski, University of Kentucky, USA
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1395:Selected Topics in Approximation and Computation,
990:2003 Arthur G. Werschulz, Fordham University, USA
1327:Iterative Methods for the Solution of Equations,
1266:2014 Bernd Carl, Aicke Hinrichs, Philipp Rudolph
1261:Joos Heintz, Bart Kuijpers, Andrés Rojas Paredes
951:Iterative Methods for the Solution of Equations.
806:time units. Even if the error is not small, say
54:but its sources remain unclear because it lacks
1404:Cambridge University Press, Cambridge, UK, 1996
1402:Noisy Information and Computational Complexity,
1208:Martin Avendano, Teresa Krick and Martin Sombra
1138:B. Bank, M. Giusti, J. Heintz, and G. M. Mbakop
1092:2011 Peter Kritzer, University of Linz, Austria
1089:2010 Daniel Rudolf, University of Jena, Germany
967:There are a number of prizes for IBC research.
1433:has a searchable data base of some 730 items.
716:(CMO) requires the calculation of a number of
167:. IBC has studied such continuous problems as
1393:Kowalski, M., Sikorski, K., and Stenger, F.,
1107:2016 Mario Hefter, TU Kaiserslautern, Germany
1083:2008 Jan Vybíral, University of Jena, Germany
1049:Winfried Sickel, University of Jena, Germany.
978:1999 Erich Novak, University of Jena, Germany
703:. See An Example: Mathematical Finance below.
8:
1416:Average-Case Analysis of Numerical Problems,
1378:: CS1 maint: multiple names: authors list (
1046:Thomas Kühn, University of Leipzig, Germany
1116:Larisa Yaroslavtseva, University of Passau
151:for the continuous problems that arise in
1425:Oxford University Press, Oxford, UK, 2001
1411:Oxford University Press, Oxford, UK, 1998
1397:Oxford University Press, Oxford, UK, 1995
1250:Lutz Kämmerer, Stefan Kunis, Daniel Potts
913:
902:Faster Valuation of Financial Derivatives
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555:Assume we want to compute an integral in
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287:
85:Learn how and when to remove this message
1423:Optimal Solution of Nonlinear Equations,
1086:2009 Steffen Dereich, TU Berlin, Germany
1390:Oxford University Press, New York, 1991
1334:A General Theory of Optimal Algorithms,
1216:2008 Stefan Heinrich and Bernhard Milla
1197:Josef Dick and Friedrich Pillichshammer
1127:Best Paper Award, Journal of Complexity
957:A General Theory of Optimal Algorithms.
1371:
329:{\displaystyle \int _{0}^{1}f(x)\,dx.}
1341:Information, Uncertainty, Complexity,
1247:Dmitriy Bilyk, V.N. Temlyakov, Rui Yu
645:. We say the problem is intractable.
345:For most integrands we can't use the
7:
1407:Traub, J. F., and Werschulz, A. G.,
1332:Traub, J. F., and Woźniakowski, H.,
1239:Leszek Plaskota, Greg W. Wasilkowski
888:, Woźniakowski) discovered that the
1283:2017 Martijn Baartse and Klaus Meer
221:, economics, mathematical finance,
1163:Bernard Mourrain and Victor Y. Pan
838:{\displaystyle \epsilon =10^{-2},}
714:collateralized mortgage obligation
25:
1202:2006 Knut Petras and Klaus Ritter
1113:Takashi Goda, University of Tokyo
1292:Julian Grote and Christoph Thäle
708:An example: mathematical finance
113:Oracle complexity (optimization)
101:
31:
1473:Computational complexity theory
1418:Springer-Verlag, New York, 2000
605:{\displaystyle \epsilon ^{-d}.}
347:fundamental theorem of calculus
177:ordinary differential equations
1343:Addison-Wesley, New York, 1983
1336:Academic Press, New York, 1980
799:{\displaystyle \epsilon ^{-d}}
756:being the number of months in
635:The exponential dependence on
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439:
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395:
389:
313:
307:
173:partial differential equations
1:
1358:. New York: Academic Press.
1356:Information-Based Complexity
263:continuous quantum computing
191:, and very-high-dimensional
137:Information-based complexity
18:Information based complexity
1409:Complexity and Information,
736:dimensional integrals, the
545:travelling salesman problem
110:It has been suggested that
1489:
1448:Complexity and Information
1141:R. DeVore and V. Temlyakov
506:Complexity and Information
894:low discrepancy sequences
689:{\displaystyle \epsilon }
669:{\displaystyle \epsilon }
572:{\displaystyle \epsilon }
532:{\displaystyle \epsilon }
127:Proposed since June 2024.
1157:1999 Arthur G. Werschulz
865:{\displaystyle 10^{720}}
519:has to be to compute an
149:computational complexity
40:This article includes a
642:curse of dimensionality
69:more precise citations.
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541:integer factorization
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493:{\displaystyle f(x).}
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165:mathematical finance
120:into this article. (
1174:2002 Peter Hertling
947:nonlinear equations
921:{\displaystyle 360}
892:(QMC) method using
878:Columbia University
749:{\displaystyle 360}
729:{\displaystyle 360}
699:We can incorporate
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239:weather forecasting
181:nonlinear equations
1386:Werschulz, A. G.,
1132:1996 Pascal Koiran
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874:Monte Carlo method
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769:{\displaystyle 30}
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243:climate prediction
185:integral equations
143:) studies optimal
42:list of references
1183:Boleslaw Kacewicz
890:quasi-Monte Carlo
508:for particulars.
500:We combine these
448:{\displaystyle .}
362:{\displaystyle f}
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1222:2010 Co-winners
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1177:2003 Co-winners
1171:2001 Erich Novak
1166:J. Maurice Rojas
1160:2000 Co-winners
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75:October 2014
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61:Please help
53:
1431:IBC website
1346:Novak, E.,
1305: [
1236:Thomas Daun
1152:P. Kirrinis
193:integration
161:engineering
67:introducing
1320:References
1258:Shu Tezuka
884:, Paskov,
247:statistics
231:geophysics
213:by noise.
145:algorithms
1374:cite book
825:−
814:ϵ
789:−
785:ϵ
684:ϵ
664:ϵ
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588:ϵ
567:ϵ
527:ϵ
415:…
291:∫
157:economics
1467:Category
845:this is
269:Overview
543:or the
373:points
251:Hilbert
219:physics
206:partial
201:complex
122:Discuss
63:improve
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963:Prizes
255:Banach
245:, and
163:, and
118:merged
1309:]
943:Traub
898:Traub
886:Traub
48:, or
1380:link
1360:ISBN
464:The
241:and
197:real
147:and
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369:at
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